The existence of at least one positive solution to a second-order nonlinear two-point boundary value problem, is established. Combining difference methods with Brouwer fixed point and Ascol & imath;-Arzel & aacute; theorems, we get a solution as the limit of an appropriate sequence of piecewise linear interpolations. Furthermore, a priori bounds on the infinite norm of a solution and its derivatives are pointed out. Some examples are also discussed to illustrate our results.

We consider the periodic problem for a 2nd order ODE with non-invertible linear part, and mild nonlinear dissipation term. The motivation for this study is a paper by Lazer [9] We add a bounded restoring force g(u) and show that the sufficient condition (of Landesman-Lazer type) given in [9] still implies the existence of a periodic solution in our case. We also comment on some variants of the problem and on the existence of bounded solutions. Ffor more information see https://ejde.math.txstate.edu/special/01/s1/abstr.html

We deal with the existence of nonnegative and nontrivial T-periodic solutions for the equation x '' = r (t)x(alpha) - s(t)x(beta) where r and s are continuous T-periodic functions and 0 < alpha < beta < 1. This equation has been studied in connection with the valveless pumping phenomenon and we will take advantage of its variational structure in order to guarantee its solvability by means of the mountain pass theorem of Ambrosetti and Rabinowitz.

We establish necessary and sufficient conditions for the existence of periodic solutions to second-order nonlinear difference equations of the form ∆xi + λxi + ∆ f (xi) = ei, i ∈N, and for a simpler equation with difference-free nonlinearity. The linear part of the equation has two-dimensional kernel.

Abstract We study some conditions of solvability of a semilinear system in ℝn, where the linear part is represented by an n × n matrix with one-dimensional kernel and the nonlinear term is a sublinear, continuous vector field.

We study a nonlinear singular boundary value problem and prove that, depending on a relationship between exponents of power terms, the problem has either solutions of Dirichlet type or homoclinic solutions. We make use of shooting techniques and lower and upper solutions.

Functional differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some varying quantities and their rates of change in space and/or time (expressed as derivatives or differences) is known or postulated.This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies.In some cases, this differential equation (called an equation of motion) may be solved explicitly.In fact, differential equations play an important role in modelling virtually every physical, technical, biological, ecological, and epidemiological process, from celestial motion, to bridge design, to interactions between neurons, to interaction between species, to spread of diseases with a population, and so forth.Also many fundamental laws of chemistry can be formulated as differential equations and in economy differential equations are used to model the behavior of complex systems.However, the mathematical models can also take different forms depending on the time scale and space structure of the problem; it can be modeled by delay differential equations, difference equations, partial delay differential equations, partial delay difference equations, or the combination of these equations.When necessary, random effects and sudden effects can also be considered in modelling problems.The mathematical theory of differential equations first developed, together with the sciences, where the equations had originated and where the results found applications.Differential and difference

Tambien esto pasara va en la octava edicion desde que fue publicado en 2015. Se cuenta que antes de salir a la venta, la novela de Milena Busquets ya habia sido vendida a, por lo menos, treinta y tres editoriales en el mundo. Arraso en la Feria de Frankfurt suscitando el interes de editoriales como Gallimard, Suhrkamp, Rizzoli o Hogarth Press de Reino Unido.

By means of variational methods we prove the existence of a positive, homoclinic solution to an equation of the kind u '' = au - bu(P), where p > 1, and both coefficients a(x), b(x) are positive and asymptotically constant. Our main result requires a control from above on the ratios M/alpha and beta/v, where M = sup a, alpha = a(infinity), v = inf b, beta= b(infinity).

We study the existence of heteroclinics connecting the two equilibria +/- 1 of the third order differential equationU ''' = f(u) + p(t)u'where f is a continuous function such that f(u)(u(2) - 1) > 0 if u not equal +/- 1 and p is a bounded non negative function. Uniqueness is also addressed.

We study the existence of monotone traveling waves \(u(t, x)=u(x+ct)\), connecting two equilibria, for the reaction-diffusion PDE \(u_{t} = (\frac{u_{x}}{\sqrt{1+u_{x}^{2}}} )_{x} + f(u)\). Assuming different forms for the reaction term \(f(u)\) (among which we have the so-called types A, B, and C), we show that, concerning the admissible speeds, the situation presents both similarities and differences with respect to the classical case. We use a first order model obtained after a suitable change of variables. The model contains a singularity and therefore has some features which are not present in the case of linear diffusion. The technique used involves essentially shooting arguments and lower and upper solutions. Some numerical simulations are provided in order to better understand the features of the model.

Let f be a continuous function in [0,1] with f(0)=0=f(1) and f>0 on ]0,1[. We show that, under additional mild conditions on f, the minimal speed for travelling waves of (0.1)∂u∂t=∂∂x[|∂u∂x|p−2∂u∂x]+f(u), may be computed via a constrained minimum problem which in turn is related to the solution of a singular boundary value problem in the half line.

It is now almost exactly 14 years that Miguel Ramos and myself were flying together to Santiago (Chile) to attend a workshop on Nonlinear Analysis at Valparaíso. I had then begun to be interested in ODEs related to the Fisher-Kolmogorov equations. During a discussion on the subjects of our talks at Valparaíso, a hint of Miguel drew my attention to aspects of the theory (where a concept of “minimal speed"plays a crucial role) for which there should be a variational approach. The mathematical content of my talk has been suggested by this memory. I will give an account of some of those variational arguments.

We study some properties of the monotone solutions of the boundary value problem(p(u'))' - cu' + f(u) = 0,u(-infinity) = 0, u(+infinity) = 1,where f is a continuous function, positive in (0, 1) and taking the value zero at 0 and 1, and P may be an increasing homeomorphism of (0, 1) or (0, +infinity) onto [0, +infinity). This problem arises when we look for travelling waves for the reaction diffusion equationpartial derivative u/partial derivative t = partial derivative/partial derivative x [p(partial derivative u/partial derivative x)] + f(u)with the parameter c representing the wave speed. A possible model for the nonlinear diffusion is the relativistic curvature operator p(nu)= nu/root 1-nu(2).The same ideas apply when P is given by the one- dimensional p- Laplacian P(v) = |v|(p-2)v. In this case, an advection term is also considered.We show that, as for the classical Fisher- Kolmogorov- Petrovski- Piskounov equations, there is an interval of admissible speeds c and we give characterisations of the critical speed c. We also present some examples of exact solutions. (C) 2014 Elsevier Inc. All rights reserved.

We study positive solutions y(u) for the first order differential equationy' = q(cy(1/p) - f(u))where c > 0 is a parameter, p > 1 and q > 1 are conjugate numbers and f is a continuous function in [0, 1] such that f(0) = 0 = f(1). We shall be particularly concerned with positive solutions y(u) such that y(0) = 0 = y(1). Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the p-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for p = 2.

Se um professor tem potencial para deixar marcas no percurso de vida dos seus estudantes, o reciproco tambem acontece. Certos alunos nao permitem que o professor se limite a uma prestacao acomodada: sustentam a manutencao de um nivel alto no ensino e acabam por influenciar o desempenho do proprio professor. Quando encontrei o Miguel Ramos como estudante do curso de Analise Funcional, que eu ensinava na FCUL em 1985, percebi que ele era um desses. Voltou a ser meu aluno numa disciplina de mestrado, onde continuou a revelar entusiasmo invulgar e urgencia em ultrapassar as metas usuais.